Definition of the Mass Moment of Inertia
Point Mass
Suppose we have a point mass $m$ which is rotating a distance $R$ away from a fixed axis of rotation with angular velocity $\omega$. Recall that the tangential velocity is $v = R\omega$.
In this simplified example, the angular momentum of this point mass is given by the following.
\[L = R \times p = R \times (m v) = R \times (m R \omega) = (m R^2) \omega\]Notice that $m R^2$ only depends on the geometry of the mass while $\omega$ only depends on the motion of the mass. While $\omega$ may change over time, $m R^2$ is constant for each object (at least in this idealized scenario). Therefore, it is natural to separate this term out and give it a name. We call this the mass moment of inertia or rotational inertia of the point mass.
\[L = I \omega \qquad\qquad\qquad I = m R^2\]Notice the similarity to the equation for linear momentum, $p = mv$. Analogous to mass in linear motion, the rotational inertia describes the objects intrinsic resistance to rotation. The larger the moment of inertia, the harder it is to get the object to rotate. Conversely, if an object with a large moment of inertia is already rotating, then it is difficult to stop. This video is a nice, short demonstration of the effect an object’s moment of inertia on its rotational motion.
A Collection of Point Masses
Suppose a collection of $n$ point masses $m_i$ are a distance $R_i$ from a fixed axis of rotation each with angular velocity $\omega$. From the above, we know that each object has angular momentum and rotational inertia.
\[L_i = (m_i R_i^2) \omega = I_i \omega\]Now, we define the following.
\[L_{\text{system}} = \sum_{i=1}^n L_i = \sum_{i=1}^n I_i \omega = \left ( \sum_{i=1}^n I_i \right ) \omega\]Since all point masses are moving with the same angular velocity (they are moving identically as a group), we can factor out $\omega$, which gives us the rotational inertia of the system.
\[I_{\text{system}} = \sum_{i=1}^n I_i\]This is true for any set of objects with the same angular velocity, not just point masses. This is a very useful property for “building-up” complicated shapes from a series of smaller shapes.
Rigid Body
A rigid body is an object that does not deform or change shape. Mathematically you can say that the distance between any two points within the body is always constant.
Consider a rigid body $\mathcal{G}$ rotating around a fixed axis $\omega$. We can consider any infinitesimal unit of area $dm$. Suppose its distance to the axis of rotation is $r_{axis}$. Note that this may be different than $\b{r}$, which denotes the position in space relative to a fixed origin.
This is essentially the same as a point mass, therefore its infinitesimal moment of inertia is
\[dI = r_{axis}^2 dm\]Since it is a rigid body there is no internal motion. Therefore, all parts of $\mathcal{G}$ are rotating with the same angular velocity, so the total rotational inertia is the sum of all the infinitesimal rotational inertias within the object $\mathcal{G}$ (analogous to a collection of point masses). For continuous objects, we integrate over its geometry.
\[I_{\mathcal{G}} = \int_{\mathcal{G}} dI = \int_{\mathcal{G}} r_{axis}^2 dm\]This is the expression that this series will be computing.
While this looks simple, its complexity hides in the subscript $\mathcal{G}$. This is saying we have to integrate with respect to the object’s geometry, which can be tricky to do correctly. We will see many examples in the subsequent posts.
The Inertia Tensor
The above formula only gives the moment of inertia for a fixed axis of rotation. In some applications, we want the ability to find the moment of inertia for any axis of rotation. For this, we need the inertia tensor.
\[\m{I} = \left[ \begin{array}{@{}rrr@{}} I_{xx} & - I_{xy} & - I_{xz} \\ - I_{yx} & I_{yy} & - I_{yz} \\ - I_{zx} & - I_{zy} & I_{zz} \end{array} \right ]\]The diagonals are called the moments of inertia with respect to the $x$, $y$, and $z$ axis. They are a measure of the resistance to rotation in the respective axes.
\[I_{xx} = \int_{\mathcal{G}} (y^2 + z^2) \; dm \qquad I_{yy} = \int_{\mathcal{G}} (z^2 + x^2) \; dm \qquad I_{zz} = \int_{\mathcal{G}} (x^2 + y^2) \; dm\]The other values are called the products of inertia with respect to the $xy$, $yz$, and $zx$ plane. They are a measure of the imbalance in the mass distribution in the respective planes. I discuss the products of inertia more in a future post.
\[I_{xy} = I_{yx} = \int_{\mathcal{G}} xy \; dm \qquad I_{yz} = I_{zy} = \int_{\mathcal{G}} yz \; dm \qquad I_{zx} = I_{xz} = \int_{\mathcal{G}} zx \; dm\]Notice that due to these definitions, the inertia tensor is always symmetric.
Calculating the Moment of Inertia for an Arbitrary Axis
Given the inertia tensor and any axis of rotation $\b{\omega}$, we can find the object’s inertia about this axis as follows
\[I_{\b{\omega}} = \u{\omega}^T \; \m{I} \; \u{\omega}\]I derive the inertia tensor and discuss this more in the next post. However, if you want you can just take it as a matter of definition.